3*(-2*x+1)^2<3 — step-by-step solution

1) Let's solve in

ℝ

3 \cdot {(- 2 x + 1)}^{2} < 3

Since

3 > 0

, divide both sides by

3

without reversing the inequality:

{(- 2 x + 1)}^{2} < 1

{(- 2 x + 1)}^{2} - 1 < 0

For

{(- 2 x + 1)}^{2} - 1

, we recognize a difference of two squares:

{(- 2 x + 1)}^{2} - 1^{2}

. Apply

a^{2} - b^{2} = (a - b) \cdot (a + b)

, with

a = - 2 x + 1

and

b = 1

You can factorize the expression:

{(- 2 x + 1)}^{2} - 1

into:

((- 2 x + 1) - 1) \cdot ((- 2 x + 1) + 1)

2) Let's solve in

ℝ

((- 2 x + 1) - 1) \cdot ((- 2 x + 1) + 1) < 0

ⓘ Parentheses are computed first:

1 - 1

0

1 + 1

2

(- 2 x + 0) \cdot (- 2 x + 2) < 0

ⓘ Adding 0 changes nothing:

- 2 x + 0

- 2 x

- 2 x \cdot (- 2 x + 2) < 0

We study the sign of each factor of the product

⇥The expression

- 2 x

is zero at:

⇥

x = 0

⇥Its coefficient of

x

-2

, being negative, it is positive before and negative after

0

⇥The expression

- 2 x + 2

is zero at:

⇥

x = \frac{-2}{-2}

⇥ⓘ The minus signs of the numerator and the denominator cancel out:

\frac{-2}{-2}

\frac{2}{2}

⇥

x = \frac{2}{2}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{2}{2}

1

⇥

x = 1

⇥Its coefficient of

x

-2

, being negative, it is positive before and negative after

1

3) Let's draw up the sign table:

▶Solution obtained:

x \in (0, 1)